If you read v13 it says the Lord took them away. For me I always assumed He took them away from the Nephite people. I didn't take that to mean that they were taken from the world entirely. I just think the Nephite/Lamanite people had become so wicked that their faith was so little that the 3 Nephites couldn't work among them and influence them. 
 
If you go back to 3 Nephi 28 it says: 
 
26 But behold, I have seen them, and they have ministered unto me. 
 
27 And behold they will be among the Gentiles, and the Gentiles shall know them not. 
 
28 They will also be among the Jews, and the Jews shall know them not. 
 
29 And it shall come to pass, when the Lord seeth fit in his wisdom that they shall minister unto all the scattered tribes of Israel, and unto all nations, kindreds, tongues and people, and shall bring out of them unto Jesus many souls, that their desire may be fulfilled, and also because of the convincing power of God which is in them. 
 
30 And they are as the angels of God, and if they shall pray unto the Father in the name of Jesus they can show themselves unto whatsoever man it seemeth them good. 
 
31 Therefore, great and marvelous works shall be wrought by them, before the great and coming day when all people must surely stand before the judgment-seat of Christ; 
 
32 Yea even among the Gentiles shall there be a great and marvelous work wrought by them, before that judgment day. 
 
It is clear from these verses that they will be working among the people of the earth in the latter-days. Their purpose will be to enhance and further the work of the Lord. They will be working among those who are good people and are searching for the truth. In that kind of environment they can be successful and the Lord will make them available to those of faith. 
 
Anyway - just my thoughts.

Thank you for the scriptures than foretell that the Three Nephites are sent back to the Earth in the latter days. This helped me understand this much better!